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Wize AP Calculus (AB) Textbook > Differential Equations

Separable Differential Equations

Basics of Differential EquationsPrevious Section
Applications of DENext Section
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Separable (First Order) Differential Equations

Key Identifiers

Separable first order DEs can be put in the form dydx=f(x)⋅g(y)\frac{dy}{dx}=f\left(x\right)\cdot g\left(y\right)

Strategies for Solving Separable DEs


IVP

If an initial condition is given, use it to solve for the unknown constant C.

Practice Question

Solve the DE dPdt=tt2−1PeP\frac{dP}{dt}=\frac{t\sqrt{t^2-1}}{Pe^P}.

Practice Question

Suppose that we know x2y′=yx^2y' = y and y(1)=1y(1)=1. Determine the equation of yy.

Practice Question

Given that dydx=4x3(y−1)\frac{dy}{dx}=4x^3\left(y-1\right) and y(0)=2y\left(0\right)=2, find y(1)y\left(1\right).

Practice Question

Given that dydx=ey−3x\frac{dy}{dx}=e^{y-3x} and y(0)=ln⁡3y\left(0\right)=\ln3, find y(ln⁡2)y\left(\ln2\right).

Hint: you will need to simplify your answer using ln⁡\ln rules
  • r ln⁡x=ln⁡(xr)r\ \ln x=\ln\left(x^r\right)
  • eln⁡x=xe^{\ln x}=x

Practice Question

Solve dydx=4e5xy\frac{dy}{dx}=\frac{4e^{5x}}{y}